import numpy as np
import pandas as pd
from num2words import num2words

# sklearn
from sklearn import metrics
from sklearn.model_selection import train_test_split
from sklearn.datasets import make_moons, make_circles, make_classification
from sklearn.metrics import mean_squared_error, mean_absolute_error

# Pytorch
import torch
from torch.autograd import Variable
import torchvision.transforms as transforms

# Visualisation libraries

## Text
from colorama import Fore, Back, Style
from IPython.display import Image, display, Markdown, Latex, clear_output

## progressbar
import progressbar

## plotly
from plotly.offline import init_notebook_mode, iplot 
import plotly.graph_objs as go
import plotly.offline as py
from plotly.subplots import make_subplots
import plotly.express as px

## seaborn
import seaborn as sns

## matplotlib
import matplotlib.pyplot as plt
from matplotlib.patches import Ellipse, Polygon
from matplotlib.font_manager import FontProperties
import matplotlib.colors as mcolors
from matplotlib import cm
plt.style.use('seaborn-whitegrid')
plt.rcParams['axes.labelsize'] = 14
plt.rcParams['xtick.labelsize'] = 12
plt.rcParams['ytick.labelsize'] = 12
plt.rcParams['text.color'] = 'k'
%matplotlib inline

import warnings
warnings.filterwarnings("ignore")

Home / Financial Analysis and Modeling

PyTorch

• Getting Started
• Logistic Regression
• Multi-layer Perceptron (MLP) for Binary Classification
• Multi-layer Perceptron (MLP) for Multi-Class Classification

Artificial Neural Networks

Artificial neural networks (ANN), also know as connectionist systems, are computing system that is inspired by animal brain neural networks. ANNs are composed of artificial neurons. These neurons are connected using connections. For a given neuron, there are multiple input and output connections.

Artificial neural network components:

• Input Layer
• Hidden (Computation) Layers
• Output Layer

There is also an element of (machine) learning. Roughly speaking, learning consists of two parts forward propagation and Backpropagation. Together with a cost function, the algorithm runs iteratively with the object of decreasing Loss and increasing the accuracy.

In this article, we develop a Pytorch Artificial Neural Network model. The number of layers can be adjusted, of course. There are a large number of resources on choosing the right number of hidden layers such as [1]. Generally speaking, we do not recommend adding too many hidden layers. For a large number of problems, one hidden layer is sufficient.

Dataset

a random n-class classification dataset can be generated using sklearn.datasets.make_classification. Here, we generate a dataset with two features and 1000 instances. Moreover, the dataset is generated for multiclass classification with five classes.

n_features =2
n_classes = 2
X, y = make_classification(n_samples = int(1e3), n_features = n_features, n_redundant=0, n_classes = n_classes,
                           n_informative=2, random_state=1, n_clusters_per_class=1)
Labels_dict = dict(zip(list(np.unique(y)), [num2words(x).title() for x in np.unique(y)]))

Data = pd.DataFrame(data = X, columns = ['Feature %i' % (i+1) for i in range(n_features)])
Target = 'Outcome Variable'
Data[Target] = y
display(Data)

def Plot_Data(X, y, ax = None):
    # adding margins
    lims = float('%1.e' % X.min())-PD['pad'], float('%1.e' % X.max())+PD['pad']   
    # Figure
    if ax == None:
        fig, ax = plt.subplots(1, 1, figsize=(PD['FigSize'], PD['FigSize']))

    scatter = ax.scatter(X[:,0], X[:,1], s=PD['cricle_size'],
                         c=y, edgecolor = 'Navy', alpha = PD['alpha'], cmap = PD['ColorMap'])
    _ = ax.legend(handles=scatter.legend_elements()[0], labels= PD['Labels'],
                  fancybox=True, framealpha=1, shadow=True, borderpad=PD['BP'], loc='best', fontsize = 14)
    _ = ax.set_xlim(lims)
    _ = ax.set_ylim(lims)
    _ = ax.set_xlabel('Feature 1')
    _ = ax.set_ylabel('Feature 2')
    _ = ax.set_aspect(1)
    _ = ax.grid(PD['grid'])
    
PD = dict(BP = .5, alpha=.7, bg_alpha = 0.25, grid = True, cricle_size = 50,
          FigSize = 7, h=0.02, pad=1, ColorMap =  'Set1', Labels = list(Labels_dict.values()))

Plot_Data(X, y, ax = None)

	Feature 1	Feature 2	Outcome Variable
0	1.536830	-1.398694	1
1	1.369176	-0.637344	1
2	0.502318	-0.459105	1
3	1.833193	-1.298082	1
4	1.042356	1.121529	0
...	...	...	...
995	0.535224	0.435245	1
996	1.069692	-0.129909	1
997	1.820267	-2.957167	1
998	1.004999	0.936290	0
999	1.462110	1.144978	0

1000 rows × 3 columns

Train and Test Sets

def DatasetTargetDist(Inp, Target, Labels_dict, PD):
    # Table
    Table = Inp[Target].value_counts().to_frame('Count').reset_index(drop = False).rename(columns = {'index':Target})
    Table[Target] = Table[Target].replace(Labels_dict)
    Table['Percentage'] = np.round(100*(Table['Count']/Table['Count'].sum()),2)
    fig = make_subplots(rows=1, cols=2, horizontal_spacing = 0.02, column_widths=PD['column_widths'],
                        specs=[[{"type": "table"},{"type": "pie"}]])

    # Right
    fig.add_trace(go.Pie(labels=Table[Target].values, values=Table['Count'].values,
                         pull=PD['pull'], textfont=dict(size= PD['textfont']),
                         marker=dict(colors = PD['PieColors'], line=dict(color='black', width=1))), row=1, col=2)
    fig.update_traces(hole=PD['hole'])
    fig.update_layout(height = PD['height'], legend=dict(orientation="v"), legend_title_text= PD['legend_title'])
    
    # Left
    T = Table.copy()
    T['Percentage'] = T['Percentage'].map(lambda x: '%%%.2f' % x)
    Temp = []
    for i in T.columns:
        Temp.append(T.loc[:,i].values)

    fig.add_trace(go.Table(header=dict(values = list(Table.columns), line_color='darkslategray',
                                       fill_color= PD['TableColors'][0], align=['center','center'],
                                       font=dict(color='white', size=12), height=25), columnwidth = PD['tablecolumnwidth'],
                           cells=dict(values=Temp, line_color='darkslategray',
                                      fill=dict(color= [PD['TableColors'][1], PD['TableColors'][1]]),
                                  align=['center', 'center'], font_size=12, height=20)), 1, 1)

    fig.update_layout(title={'text': '<b>' + Target + '<b>', 'x':PD['title_x'],
                                     'y':PD['title_y'], 'xanchor': 'center', 'yanchor': 'top'})
    fig.show()

PD = dict(PieColors = ['OrangeRed', 'RoyalBlue'], TableColors = ['Navy','White'], hole = .4,
          column_widths=[0.6, 0.4],textfont = 14, height = 350, tablecolumnwidth = [0.32, 0.15, 0.15],
          pull = [0.05], legend_title = Target, title_x = 0.5, title_y = 0.8)
DatasetTargetDist(Data, Target, Labels_dict, PD)

Modeling: PyTorch Multi-layer Perceptron (MLP) for Binary Classification

A multi-layer perceptron (MLP) is a class of feedforward artificial neural network (ANN). The algorithm at each iteration uses the Cross-Entropy Loss to measure the loss, and then the gradient and the model update is calculated. At the end of this iterative process, we would reach a better level of agreement between test and predicted sets since the error would be lower from that of the first step.

Train and Test sets

One of the efficient methods of splitting a dataset into random train and test subsets is using sklearn.model_selection.train_test_split.

X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.3, random_state=42)

pd.DataFrame(data={'Set':['X_train','X_test','y_train','y_test'],
               'Shape':[X_train.shape, X_test.shape, y_train.shape, y_test.shape]}).set_index('Set').T

Set	X_train	X_test	y_train	y_test
Shape	(700, 2)	(300, 2)	(700,)	(300,)

def Header(Text, L = 100, C = 'Blue', T = 'White'):
    BACK = {'Black': Back.BLACK, 'Red':Back.RED, 'Green':Back.GREEN, 'Yellow': Back.YELLOW, 'Blue': Back.BLUE,
         'Magenta':Back.MAGENTA, 'Cyan': Back.CYAN}
    FORE = {'Black': Fore.BLACK, 'Red':Fore.RED, 'Green':Fore.GREEN, 'Yellow':Fore.YELLOW, 'Blue':Fore.BLUE,
         'Magenta':Fore.MAGENTA, 'Cyan':Fore.CYAN, 'White': Fore.WHITE}
    print(BACK[C] + FORE[T] + Style.NORMAL + Text + Style.RESET_ALL + ' ' + FORE[C] +
          Style.NORMAL +  (L- len(Text) - 1)*'=' + Style.RESET_ALL)
    
def Line(L=100, C = 'Blue'):
    FORE = {'Black': Fore.BLACK, 'Red':Fore.RED, 'Green':Fore.GREEN, 'Yellow':Fore.YELLOW, 'Blue':Fore.BLUE,
         'Magenta':Fore.MAGENTA, 'Cyan':Fore.CYAN, 'White': Fore.WHITE}
    print(FORE[C] + Style.NORMAL + L*'=' + Style.RESET_ALL)
    
def Search_List(Key, List): return [s for s in List if Key in s]

def TorchSets(Set):
    # Inut: Arrays
    # GPU Cuda
    if isinstance(Set, (pd.DataFrame, pd.Series)):
        Set = Set.values
    if torch.cuda.is_available():
        if Set.ndim==1:
            Out = Variable(torch.from_numpy(Set).type(torch.LongTensor).cuda())
        else:
            Out = Variable(torch.from_numpy(Set).cuda())
    # CPU
    else:
        if Set.ndim==1:
            Out = Variable(torch.from_numpy(Set).type(torch.LongTensor))
        else:
            Out = Variable(torch.from_numpy(Set))
    return Out

# Tensors
X_train_tensor = TorchSets(X_train)
y_train_tensor = TorchSets(y_train)
X_test_tensor = TorchSets(X_test)
y_test_tensor = TorchSets(y_test)
    
Batch_size = 100
iteration_number = int(3e4)

epochs_number = int(iteration_number / (len(X_train) / Batch_size))

# Pytorch train and test sets
Train_set = torch.utils.data.TensorDataset(X_train_tensor, y_train_tensor)
Test_set = torch.utils.data.TensorDataset(X_test_tensor, y_test_tensor)

# data loader
train_loader = torch.utils.data.DataLoader(Train_set, batch_size = Batch_size, shuffle = False)
test_loader = torch.utils.data.DataLoader(Train_set, batch_size = Batch_size, shuffle = False)

class MLP_Model(torch.nn.Module):
    '''
    A MLP model with two hidden layers
    '''
    def __init__(self, input_Size, hidden_Size, output_Size):
        super(MLP_Model, self).__init__()
        
        # Input Layer to the 1st Layer:
        self.fc1 = torch.nn.Linear(input_Size, hidden_Size) 
        torch.nn.init.kaiming_uniform_(self.fc1.weight, nonlinearity='relu')
        self.act1 = torch.nn.ReLU()
        
        # 1st Layer to 2nd Layer
        self.fc2 = torch.nn.Linear(hidden_Size, int(hidden_Size/4))
        torch.nn.init.kaiming_uniform_(self.fc2.weight, nonlinearity='relu')
        self.act2 = torch.nn.ReLU()
        
        # 2nd layer to Output Layer
        self.fc3 = torch.nn.Linear(int(hidden_Size/4), output_Size)
        torch.nn.init.kaiming_uniform_(self.fc3.weight)
        self.act3 = torch.nn.Sigmoid()
    
    def forward(self, x):
        # Input Layer to the 1st Layer:
        out = self.fc1(x)
        # Non-linearity 1
        out = self.act1(out)
        
        # 1st Layer to 2nd Layer
        out = self.fc2(out)
        out = self.act2(out)
               
        # 2nd layer to Output Layer
        out = self.fc3(out)
        out = self.act3(out)
        return out

def Plot_history(history, Table_Rows = 25, yLim = 2):
    fig = make_subplots(rows=1, cols=2, horizontal_spacing = 0.02, column_widths=[0.6, 0.4],
                        specs=[[{"type": "scatter"},{"type": "table"}]])
    # Left
    fig.add_trace(go.Scatter(x= history['Iteration'].values, y= history['Loss'].astype(float).values.round(4),
                             line=dict(color='OrangeRed', width= 1.5), name = 'Loss'), 1, 1)
    fig.add_trace(go.Scatter(x= history['Iteration'].values, y= history['Accuracy'].astype(float).values,
                             line=dict(color='MidnightBlue', width= 1.5),  name = 'Accuracy'), 1, 1)
    fig.update_layout(legend=dict(x=0, y=1.1, traceorder='reversed', font_size=12),
                  dragmode='select', plot_bgcolor= 'white', height=600, hovermode='closest',
                  legend_orientation='h')
    fig.update_xaxes(range=[history.Iteration.min(), history.Iteration.max()],
                     showgrid=True, gridwidth=1, gridcolor='Lightgray',
                     showline=True, linewidth=1, linecolor='Lightgray', mirror=True, row=1, col=1)
    fig.update_yaxes(range=[0, yLim], showgrid=True, gridwidth=1, gridcolor='Lightgray',
                     showline=True, linewidth=1, linecolor='Lightgray', mirror=True, row=1, col=1)
    # Right
    ind = np.linspace(0, history.index[-1], Table_Rows, endpoint = True).round(0).astype(int)
    T = history[history.index.isin(ind)]
    T[['Loss','Accuracy']] = T[['Loss','Accuracy']].applymap(lambda x: '%.4e' % x)
    Temp = []
    for i in T.columns:
        Temp.append(T.loc[:,i].values)
    fig.add_trace(go.Table(header=dict(values = list(history.columns), line_color='darkslategray',
                                       fill_color='Navy', align=['center','center'],
                                       font=dict(color='white', size=12), height=25), columnwidth = [0.4, 0.4, 0.4],
                           cells=dict(values=Temp, line_color='darkslategray',
                                      fill=dict(color=['Lavender', 'white', 'white']),
                                      align=['center', 'center'], font_size=12,height=20)), 1, 2)
    fig.show()

Fitting the model

input_Size, output_Size = n_features, len(Labels_dict)
hidden_Size = 256

# model
model = MLP_Model(input_Size, hidden_Size, output_Size)

# GPU
if torch.cuda.is_available():
    model.cuda()

# Cross Entropy Loss 
criterion= torch.nn.CrossEntropyLoss()

# Optimizer 
optimizer = torch.optim.SGD(model.parameters(), lr= 1e-2, momentum=.9)

# Traning the Model
Count = 0
Loss_list = []
Iteration_list = []
Accuracy_list = []
MSE_list = []
MAE_list = []
Steps = 10

Progress_Bar = progressbar.ProgressBar(maxval= iteration_number + 200,
                                       widgets=[progressbar.Bar('=', '|', '|'),
                                                progressbar.Percentage()])
    
# print('---------------------------------------------------------')
for epoch in range(epochs_number):
    for i, (Xtr, ytr) in enumerate(train_loader):
        
        # Variables
        Xtr = Variable(Xtr.view(-1, n_features))
        ytr = Variable(ytr)
        
        # Set all gradients to zero
        optimizer.zero_grad()
        
        # Forward
        Out = model(Xtr.float())
        
        # loss
        loss = criterion(Out, ytr.long())
        
        # Backward (Calculating the gradients)
        loss.backward()
        
        # Update parameters
        optimizer.step()
        
        Count += 1
        
        del Xtr, ytr
        
        # Predictions
        if Count % Steps == 0:
            # Calculate Accuracy         
            Correct, Total = 0, 0
            # Predictions
            for Xts, yts in test_loader: 
                Xts = Variable(Xts.view(-1, n_features))
                
                # Forward
                Out = model(Xts.float())
                
                # The maximum value of Out
                Predicted = torch.max(Out.data, 1)[1]
                
                # Total number of yts
                Total += len(yts)
                
                # Total Correct predictions
                Correct += (Predicted == yts).sum()
            del Xts, yts
            # storing loss and iteration
            Loss_list.append(loss.data)
            Iteration_list.append(Count)
            Accuracy_list.append(Correct / float(Total))
            
        Progress_Bar.update(Count)

Progress_Bar.finish()

history = pd.DataFrame({'Iteration': np.array(Iteration_list),
                      'Loss': np.array([x.cpu().data.numpy() for x in Loss_list]),
                      'Accuracy': np.array([x.cpu().data.numpy() for x in Accuracy_list])})
del Loss_list, Iteration_list, Accuracy_list

|=========================================================================|100%

Model Performance

Plot_history(history, Table_Rows = 30, yLim = 1)

def Plot_Classification(Model, X, y, ax = None):
    # adding margins
    x_min, x_max = float('%1.e' % X.min())-PD['pad'], float('%1.e' % X.max())+PD['pad']
    y_min, y_max = float('%1.e' % X.min())-PD['pad'], float('%1.e' % X.max())+PD['pad']
    # Generating meshgrids
    xx, yy = np.meshgrid(np.arange(x_min, x_max, PD['h']), np.arange(y_min, y_max, PD['h']))

    Temp = np.c_[xx.ravel(), yy.ravel()]
    Temp = TorchSets(Temp)

    # Predictions
    Pred = Model(Temp.float())
    Pred = torch.max(Pred.data, 1)[1]
    Pred = Pred.cpu().data.numpy()
    Pred = Pred.reshape(xx.shape)

    # Figure
    if ax == None:
        fig, ax = plt.subplots(1, 1, figsize=(PD['FigSize'], PD['FigSize']))
        
    _ = ax.contourf(xx, yy, Pred, cmap = PD['ColorMap'], alpha=PD['bg_alpha'])

    scatter = ax.scatter(X[:,0], X[:,1], s=PD['cricle_size'],
                         c=y, edgecolor = 'Navy', alpha = PD['alpha'], cmap = PD['ColorMap'])
    _ = ax.legend(handles=scatter.legend_elements()[0], labels= PD['Labels'],
                  fancybox=True, framealpha=1, shadow=True, borderpad=PD['BP'], loc='best', fontsize = 14)
    _ = ax.set_xlim(x_min, x_max)
    _ = ax.set_ylim(y_min, y_max)
    _ = ax.set_xlabel('Feature 1')
    _ = ax.set_ylabel('Feature 2')
    _ = ax.set_aspect(1)
    _ = ax.grid(PD['grid'])

#    
PD = dict(BP = .5, alpha=.7, bg_alpha = 0.25, grid = False, cricle_size = 50,
          FigSize = 7, h=0.02, pad=1, ColorMap =  'Set1', Labels = list(Labels_dict.values()))
fig, ax = plt.subplots(1, 2, figsize=(16, 7))
# Train Set
Plot_Classification(model, X_train, y_train, ax = ax[0])
_ = ax[0].set_title('Train Set', fontsize = 16)
# Test Set
Plot_Classification(model, X_test, y_test, ax = ax[1])
_ = ax[1].set_title('Test Set', fontsize = 16)

Confusion Matrix

The confusion matrix allows for visualization of the performance of an algorithm. Note that due to the size of data, here we don't provide a Cross-validation evaluation. In general, this type of evaluation is preferred.

def Confusion_Mat(CM_Train, CM_Test, PD, n_splits = 10):
    if n_splits == None:
        Titles = ['Train Set', 'Test Set']
    else:
        Titles = ['Train Set (CV = % i)' % n_splits, 'Test Set (CV = % i)' % n_splits]
    CM = [CM_Train, CM_Test]
    Cmap = ['Greens', 'YlGn','Blues', 'PuBu']
    for i in range(2):
        fig, ax = plt.subplots(1, 2, figsize= PD['FS'])
        fig.suptitle(Titles[i], weight = 'bold', fontsize = 16)
        _ = sns.heatmap(CM[i], annot=True, annot_kws={"size": PD['annot_kws']}, cmap=Cmap[2*i], ax = ax[0],
                        linewidths = 0.2, cbar_kws={"shrink": PD['shrink']})
        _ = ax[0].set_title('Confusion Matrix');
        Temp = np.round(CM[i].astype('float') / CM[i].sum(axis=1)[:, np.newaxis], 2)
        _ = sns.heatmap(Temp,
                        annot=True, annot_kws={"size": PD['annot_kws']}, cmap=Cmap[2*i+1], ax = ax[1],
                       linewidths = 0.4, vmin=0, vmax=1, cbar_kws={"shrink": PD['shrink']})
        _ = ax[1].set_title('Normalized Confusion Matrix');

        for a in ax:
            _ = a.set_xlabel('Predicted labels')
            _ = a.set_ylabel('True labels'); 
            _ = a.xaxis.set_ticklabels(PD['Labels'])
            _ = a.yaxis.set_ticklabels(PD['Labels'])
            _ = a.set_aspect(1)

# Train
y_pred = model(X_train_tensor.float())
y_pred = torch.max(y_pred.data, 1)[1]
y_pred = y_pred.cpu().data.numpy()
Reports_Train = pd.DataFrame(metrics.classification_report(y_train, y_pred, target_names=list(Labels_dict.values()),
                                                           output_dict=True)).T
CM_Train = metrics.confusion_matrix(y_train, y_pred)
# Test
y_pred = model(X_test_tensor.float())
y_pred = torch.max(y_pred.data, 1)[1]
y_pred = y_pred.cpu().data.numpy()
Reports_Test = pd.DataFrame(metrics.classification_report(y_test, y_pred, target_names=list(Labels_dict.values()),
                                                          output_dict=True)).T
CM_Test = metrics.confusion_matrix(y_test, y_pred)

Reports_Train = Reports_Train.reset_index().rename(columns ={'index': 'Train Set'})
Reports_Test = Reports_Test.reset_index().rename(columns ={'index': 'Test Set'})
                                                 
display(Reports_Train.style.hide_index().set_properties(**{'background-color': 'HoneyDew', 'color': 'Black'}).\
        set_properties(subset=['Train Set'], **{'background-color': 'SeaGreen', 'color': 'White'}))
display(Reports_Test.style.hide_index().set_properties(**{'background-color': 'Azure', 'color': 'Black'}).\
        set_properties(subset=['Test Set'], **{'background-color': 'RoyalBlue', 'color': 'White'}))

PD = dict(FS = (10, 6), annot_kws = 14, shrink = .6, Labels = list(Labels_dict.values()))
Confusion_Mat(CM_Train, CM_Test, PD = PD, n_splits = None)

Train Set	precision	recall	f1-score	support
Zero	0.942857	0.956522	0.949640	345.000000
One	0.957143	0.943662	0.950355	355.000000
accuracy	0.950000	0.950000	0.950000	0.950000
macro avg	0.950000	0.950092	0.949997	700.000000
weighted avg	0.950102	0.950000	0.950003	700.000000

Test Set	precision	recall	f1-score	support
Zero	0.947020	0.922581	0.934641	155.000000
One	0.919463	0.944828	0.931973	145.000000
accuracy	0.933333	0.933333	0.933333	0.933333
macro avg	0.933241	0.933704	0.933307	300.000000
weighted avg	0.933701	0.933333	0.933351	300.000000

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Refrences

1. Stathakis, D. (2009). How many hidden layers and nodes?. International Journal of Remote Sensing, 30(8), 2133-2147.
2. Artificial neural network. Retrieved June 02, 2020, from https://en.wikipedia.org/wiki/Artificial_neural_network.

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